An ultrasonic resonance scattering spectroscopy technique is developed and applied for reconstructing elastic constants of a transversely isotropic cylindrical component. Immersion ultrasonic measurements are performed on tube samples made from a boron/aluminum composite material to obtain resonance frequencies and dispersion curves of different guided wave modes propagating in the tube. Theoretical analysis on the acoustic resonance scattering from a transversely isotropic cylindrical tube is also performed, from which complete backscattering and resonance scattering spectra and theoretical dispersion curves are calculated. A sensitive change of the dispersion curves to the elastic properties of the composite tube is observed for both normal and oblique incidences; this is exploited for a systematic evaluation of damage and elastic constants of the composite tube samples. The elastic constants of two boron/aluminum composite tube samples manufactured under different conditions are reconstructed through an optimization procedure in which the residual between the experimental and theoretical phase velocities (dispersion curves) is minimized.